Exact or similar copies of code fragments in a code base are known as code clones. Code clones are considered as one of the serious code smells. Stability is a widely investigated perspective of assessing the impacts of clones on software systems. A number of existing studies show that clones are often less stable than non-cloned code. This suggests that clones change more frequently than non-cloned code and thus may require comparatively more maintenance efforts. Again, frequent changes to clones may increase the likelihood of missing change propagation to the co-change candidates leading to inconsistencies or bugs. However, none of the existing studies investigate whether stability of clones is related to the bug-proneness. In this paper, we present an empirical study that analyzes the relationships between stability and bug-proneness of clones. We identify bugfix commits by analyzing the commit messages from software repositories. We then identify the clones those are changed in the bug-fix commits as bug-prone clones. We then compare the stability of buggy and non-buggy clones considering the finegrained syntactic change types and their significance. Our experimental results based on five open-source Java systems of different size and application domains show that (1) stability and bug-proneness of code clones are related and this relationship is statistically significant, (2) for both exact (Type 1) and near-miss (Type 2 and Type 3) clones, buggy clones tend to have higher frequency of changes than non-buggy clones, (3) the bug-proneness of Type 2 and Type 3 clones tend to be strongly related with their stability compared to Type 1 clones, and (4) the relation between the stability and the bug-proneness of clones with respect to fine-grained change types is likely to be influenced by the changes of low to medium significance. We believe that our findings are important and potentially useful in identifying and prioritizing candidate clones for management.